A regularization approach to functional It\^o calculus and strong-viscosity solutions to path-dependent PDEs

0.55 MB·

by  Andrea Cosso

A regularization approach to functional Itˆo calculus and
strong-viscosity solutions to path-dependent PDEs
Andrea COSSO∗ Francesco RUSSO†
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January 21, 2014
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n
a
J
Abstract
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First, we revisit functional Itoˆ/path-dependent calculus started by B. Dupire, R.

ContandD.-A.Fourni´e,usingthe formulationofcalculusviaregularization. Relations
]
R
withthecorrespondingBanachspacevaluedcalculusintroducedbyC.DiGirolamiand
P
the second named author are explored. The second partof the paper is devoted to the
.
h studyofthe KolmogorovtypeequationassociatedwiththesocalledwindowBrownian
t
a motion, called path-dependent heat equation, for which well-posedness at the level of
m classicalsolutionsisestablished. Then,anotionofstrongapproximatingsolution,called
[ strong-viscosity solution, is introduced which is supposed to be a substitution tool to

theviscositysolution. Forthatkindofsolution,wealsoproveexistenceanduniqueness.
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v The notion of strong-viscosity solution motivates the last part of the paper which is
4 devotedtoexplorethisnewconceptofsolutionforgeneralsemilinearPDEsinthefinite
3
dimensionalcase. Weproveanequivalenceresultbetweentheclassicalviscositysolution
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5 and the new one. The definition of strong-viscosity solution for semilinear PDEs is
. inspired by the notion of good solution, and it is based again on an approximating
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0 procedure.
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1
:
v Key words: Horizontal and vertical derivative; functional Itˆo/path-dependent calculus;
i
X strong-viscosity solutions; calculus via regularization.
r
a
2010 Math Subject Classification: 35D35; 35D40; 35K10; 60H05; 60H10; 60H30.
LaboratoiredeProbabilit´esetMod`elesAl´eatoires,CNRS,UMR7599,Universit´eParisDiderot,France.
∗
E-mail: [email protected]
ENSTA ParisTech, Unit´e de Math´ematiques appliqu´ees, 828, boulevard des Mar´echaux, F-91120
†
Palaiseau, France. E-mail: [email protected]
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1 Introduction
Recently,anewbranchofstochasticcalculushasappeared,knownasfunctionalItoˆcalculus,
which results to be an extension of classical Itˆo calculus to functionals depending on the
all path of a stochastic process and not only on its current value, see Dupire [16], Cont
and Fourni´e [5, 6, 7]. On the other hand, C. Di Girolami, the second named author and
more recently G. Fabbri, have introduced in a series of papers ([12, 13, 14, 15, 11]), a
stochastic calculus via regularizations for processes taking values in a separable Banach
space B, which includes (when B = C([ T,0]), as applications a path-dependent type
−
calculus having similar objectives.
In the first part of the present paper, we revisit functional Itˆo calculus by means of
stochastic calculus via regularization. We recall that developed functional Itˆo calculus
and derived a functional Itˆo’s formula using discretization techniques of F¨ollmer [23] type,
instead of regularization techniques. Let us illustrate another difference with respect to
[5]. One of the main issues of functional Itˆo calculus is the definition of the functional (or
pathwise) derivatives, i.e., the horizontal derivative (calling in only the past values of the
trajectory) and the vertical derivative (calling in only the present value of the trajectory).
In [5], it is essential to consider functionals defined on the space of c`adla`g trajectories, since
the definition of functional derivatives necessitates of discontinuous paths. Therefore, if a
functional is defined only on the space of continuous trajectories (because, e.g., it depends
on the paths of a continuous process as Brownian motion), we have to extend it anyway to
the space of c`adla`g trajectories, even though, in general, there is no unique way to extend
it. In contrast to this approach, we introduce an intermediate space between the space of
continuous trajectories C([ T,0]) and the space of c`adla`g trajectories D([ T,0]), denoted
− −
C([ T,0]), which allows us to define functional derivatives. C([ T,0]) is the space of
− −
bounded trajectories on [ T,0], continuous on [ T,0[ and with possibly a jump at 0. We
− −
endow C([ T,0]) with a topology such that C([ T,0]) is dense in C([ T,0]) with respect
− − −
to this topology. Therefore, any functional : [0,T] C([ T,0]) R, continuous with
U × − →
respect to the topology of C([ T,0]), admits a unique extension to C([ T,0]), denoted
− −
u: [0,T] C([ T,0]) R. We present some significant functionals for which a continuous
× − →
extensionexists. Then,wedevelopthefunctionalItˆocalculusforu: [0,T] C([ T,0]) R.
× − →
Notice that we use a slightly different notation with respect to [5]. In particular, in
place of a map : [0,T] C([ T,0]) R, in [5] a family of maps F = (F ) , with
t t∈[0,T]
U × − →
F : C([0,t]) R, is considered. However, we can always move from one formulation to
t
→
the other. Indeed, given F = (F ) , where each F : C([0,t]) R, we can define
t t∈[0,T] t
→
: [0,T] C([ T,0]) R as follows:
U × − →
(t,η) := F (η( +T) ), (t,η) [0,T] C([ T,0]).
t [0,t]
U · | ∈ × −
Vice-versa, let : [0,T] C([ T,0]) R and define F = (F ) as
t t∈[0,T]
U × − →
F (η˜) := (t,η), (t,η˜) [0,T] C([0,t]), (1.1)
t
U ∈ ×
where η is the element of C([ T,0]) obtained from η˜ firstly translating η˜ on the interval
−
[ t,0], then extending it in a constant way up to T, namely η(x) := η˜(x+t)1 (x)+
[−t,0]
− −
2
η˜( t)1 (x), for any x [ T,0]. Observe that, in principle, the map contains
[−T,−t)
− ∈ − U
more information than F, since in (1.1) we do not take into account the values of at
U
(t,η) [0,T] C([ T,0]) with η not constant on the interval [ T, t]. Despite this, the
∈ × − − −
equivalencebetweenthetwonotationsisguaranteedbythefactthat,asitwillbeclearlater,
when we consider the composition of with a stochastic process, this extra information
U
plays no role. Our formulation has two advantages. Firstly, we can work with a single
map instead of a family of maps. In addition, the time variable and the path have two
distinct roles in our setting, as for the time variable and the space variable in the classical
Itˆo calculus. This, in particular, allows us to define the horizontal derivative independently
of the time derivative, so that, the horizontal derivative defined in [5] corresponds to the
sum of our horizontal derivative and of the time derivative. We mention that an alternative
approach to functional derivatives was introduced in [1].
We end the first part of the paper showing how our functional Itˆo’s formula is strictly
related to the Itˆo’s formula derived in the framework of Banach space valued stochastic
calculus via regularization, for the case of window processes. This latter and brand new
branch of stochastic calculus and stochastic analysis has been recently conceived, deeply
studied, and developed in many directions in [15, 14, 13], [11] and for more details [12]. For
the particular case of window processes, we also refer to Theorem 6.3 and Section 7.2 in
[11]. In the present paper, we prove formulae which allow to express functional derivatives
in terms of differential operators arising in the Banach space valued stochastic calculus
via regularization, with the aim of identifying the building blocks of our functional Itˆo’s
formula with the terms appearing in the Itˆo’s formula for window processes.
Dupire [16] introduced also the concept of path-dependent partial differential equation,
to which the second part of the present paper is devoted. Di Girolami and the second
named author, in Chapter 9 of [12], considered a similar equation in the framework of
Banach space valued calculus, for which we refer also to [22]. We focus on path-dependent
nonlinear Kolmogorov equations driven by the path-dependent heat operator, for which
we provide a definition of classical solution. We prove a uniqueness result for this kind of
solution, by means of probabilistic methods based on the theory of backward stochastic
differential equations (BSDEs). More precisely, we show that, if a classical solution exists,
then it can be expressed through the solution of a certain backward stochastic differential
equation. Therefore, from the uniqueness of the BSDE it follows that there exists at most
one classical solution. Then, we prove an existence result for classical solutions. However,
this notion of solution turns out to be unsuitable to deal with all significant examples. As
a matter of fact, if we consider the path-dependent PDE arising in the hedging problem
of lookback contingent claims, we can not expect too much regularity of the solution (this
example is studied in detail in subsection 3.2). Therefore, we are led to consider a weaker
notion of solution to the path-dependentnonlinear Kolmogorov equation. In particular, we
are interested in a viscosity-type solution, namely a solution which is not required to be
differentiable, but only locally uniformly continuous.
The issue of providing a suitable definition of viscosity solutions for path-dependent
PDEs has attracted a great interest. We recall that Ekren, Keller, Touzi, and Zhang [17]
and Ekren, Touzi, and Zhang [19, 20] recently provided a definition of viscosity solution to
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path-dependentPDEs, replacingtheclassical minimum/maximumproperty, whichappears
in the standard definition of viscosity solution, with an optimal stopping problem under
nonlinear expectation [18]. We also recall that other definitions of viscosity solutions for
path-dependent PDEs were given by Peng [33] and Tang and Zhang [42]. In contrast with
the above cited papers, our definition of solution is not inspired by the standard definition
of viscosity solution given in terms of test functions or jets. In fact, our weak solution,
called strong-viscosity solution, to the path-dependent nonlinear Kolmogorov equation is
defined, in a few words, as the pointwise limit of classical solutions to perturbed equations.
We notice that our definition is more similar in spirit to the concept of good solution,
which turned out to be equivalent to the definition of Lp-viscosity solution for certain fully
nonlinear partial differential equations, see, e.g., [3], [9], [26], and [27]. It has also some
similarities with thevanishing viscosity method,whichrepresents oneoftheprimitiveideas
leading to the conception of the modern definition of viscosity solution. Our definition is
likewise inspired by the notion of strong solution (which justifies the name of our solution),
as defined for example in [2], [24], and [25], even though strong solutions are required to be
more regular than simply locally uniformly continuous (this regularity is usually required
to prove uniqueness of strong solutions, which for example in [24] and [25] is based on a
Fukushima-Dirichlet decomposition). Instead, our definition of strong-viscosity solution to
the path-dependentnonlinear Kolmogorov equation is only required to belocally uniformly
continuous and with polynomial growth. The term viscosity in the name of our solution is
also justifies by the fact that in the finite dimensional case we have an equivalence result
between the notion of strong-viscosity solution and that of viscosity solution. We prove a
uniqueness theorem for strong-viscosity solutions using the theory of backward stochastic
differential equations and we provide an existence result.
We conclude the second part of the paper analyzing more in detail the notion of strong-
viscosity solution for semilinear equations, focusing on the more understandable finite di-
mensional case. In particular, we notice that, if from one hand we consider only equations,
as well as perturbed equations, whose classical solutions admit a nonlinear Feynman-Kac
representation formula in terms of BSDEs, on the other hand our definition of strong-
viscosity solution has the advantage that the comparison theorem follows directly from
the comparison theorem for BSDEs. In other words, the comparison theorem for strong-
viscosity solutions can be proved using probabilistic methods, in contrast to real analysis’
tools which characterize comparison theorems for viscosity solutions and revealed to be ar-
duoustoextendtotheinfinitedimensionalsetting,see,e.g.,[21]. Wepresenttwodefinitions
of strong-viscosity solution, one of them is more in the spirit of the standard definition of
viscosity solution,sinceitisrequiredtobebothastrong-viscosity subsolutionandastrong-
viscosity supersolution. A strong-viscosity supersolution (resp. subsolution) is defined, in
few words, to be the pointwise limit of classical supersolutions (resp. subsolutions) to per-
turbed semilinear equations. We prove, usingthe theory of backward stochastic differential
equations, that a comparison theorem for strong-viscosity sub and supersolutions holds,
therefore obtaining a uniqueness result for our strong-viscosity solution. More precisely, we
prove that every strong-viscosity supersolution (resp. subsolution) can be represented by a
supersolution (resp. subsolution) of a BSDE. Indeed, every strong-viscosity supersolution
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is the limit of a sequence of classical supersolutions, which admit a representation in terms
of supersolutionsof BSDEs. Then, usingalimit theorem for BSDEs (partly inspiredby the
monotonic limit theorem of Peng [32]), we derive a limit BSDE supersolution, which turns
out to be a probabilistic representation for our strong-viscosity supersolution. Therefore,
as already mentioned, the comparison theorem for strong-viscosity sub and supersolutions
is a consequence of the comparison theorem for BSDEs. We conclude investigating the
equivalence between the notion of strong-viscosity solution and the standard definition of
viscosity solution.
The rest of the paper is organized as follows. In section 2 we develop functional Itˆo cal-
culus via regularization: after a brief introduction on finite dimensional stochastic calculus
via regularization in subsection 2.1, we introduce and study the space C([ T,0]) in subsec-
−
tion 2.2; then, we define the pathwise derivatives and we prove the functional Itˆo’s formula
in subsection 2.3; in subsection 2.4, instead, we discuss the relation between functional Itˆo
calculus via regularization and Banach space valued stochastic calculus via regularization
for window processes. In section 3, on the other hand, we study path-dependent PDEs.
More precisely, in subsection 3.1 we discuss classical solutions to the path-dependent non-
linear Kolmogorov equation; in subsection 3.2 we present a significant hedging example to
motivate the introduction of a weaker notion of solution; in subsection 3.3 we provide the
definition of strong-viscosity solution to the path-dependent nonlinear Kolmogorov equa-
tion; finally, in subsection 3.4 we explore the notion of strong-viscosity solution for more
general PDEs (not only PDEs driven by the heat operator), in the finite dimensional case.
2 Functional Itoˆ calculus: a regularization approach
2.1 Background: finite dimensional calculus via regularization
The theory of stochastic calculus via regularization has been developed in several papers,
starting from [37, 38]. We recall below only the results used in the present paper, and we
refer to [40] for a survey on the subject. We emphasize that integrands are allowed to be
anticipating. Moreover, the integration theory and calculus appears to be close to a pure
pathwise approach even though there is still a probability space behind.
Fix a probability space (Ω, ,P) and T ]0, [. Let F = ( ) denote a filtration
t t∈[0,T]
F ∈ ∞ F
satisfying the usual conditions. Let X = (X ) (resp. Y = (Y ) ) be a real
t t∈[0,T] t t∈[0,T]
continuous (resp. P-a.s. integrable) process. Every real continuous process X = (X )
t t∈[0,T]
is naturally extended to all t R setting X = X , t 0, and X = X , t T. We also
t 0 t T
∈ ≤ ≥
define a C([ T,0])-valued process X = (X ) , called the window process associated
t t∈R
−
with X, defined by
X := X , x [ T,0] , t R.
t t+x
{ ∈ − } ∈
Definition 2.1 Suppose that, for every t [0,T], the following limit
∈
t t X X
Y d−X := lim Y s+ε− sds, (2.1)
s s s
Z0 ε→0+Z0 ε
5
exists in probability. If the obtained random function admits a continuous modification,
that process is denoted by ·Yd−X and called forward integral of Y with respect to
0
X.
R
(ε)
Definition 2.2 A family of processes (H ) is said to converge to (H ) in the
t t∈[0,T] t t∈[0,T]
ucp sense, if sup H(ε) H goes to 0 in probability, as ε 0+.
0≤t≤T | t − t| →
Proposition 2.1 Suppose that the limit (2.1) exists in the ucp sense. Then, the forward
integral ·Yd−X of Y with respect to X exists.
0
Let uRs introduce the concept of covariation, which is a crucial notion in stochastic
calculus via regularization. Let us suppose that X,Y are continuous processes.
Definition 2.3 The covariation of X and Y is defined by
1 t
[X,Y] = [Y,X] = lim (X X )(Y Y )ds, t [0,T],
t t ε→0+ ε Z0 s+ε− s s+ε− s ∈
if the limit exists in probability for every t [0,T], provided that the limiting random
∈
function admits a continuous version (this is the case if the limit holds in the ucp sense). If
X = Y, X is said to be a finite quadratic variation process and we set [X] := [X,X].
The forward integral and the covariation generalize the classical Itˆo integral and covari-
ation for semimartingales. In particular, we have the following result, for a proof we refer
to, e.g., [40].
Proposition 2.2 The following properties hold:
(i) Let S1,S2 be continuous F-semimartingales. Then, [S1,S2] is the classical bracket
[S1,S2] = M1,M2 , where M1 (resp. M2) is the local martingale part of S1 (resp.
h i
S2).
(ii) Let V be a continuous bounded variation process and Y be a ca`dl`ag process (or vice-
versa); then [V] = [Y,V] = 0. Moreover ·Yd−V = ·YdV, is the Lebesgue-
0 0
Stieltjes integral.
R R
(iii) If W is a Brownian motion and Y is an F-progressively measurable process such that
T Y2ds < , P-a.s., then ·Yd−W exists and equals the Itoˆ integral ·YdW.
0 s ∞ 0 0
WeR end this crash introductioRn to finite dimensional stochastic calculusR via regular-
ization presenting one of its cornerstones: Itˆo’s formula. It is a well-known result in the
theory of semimartingales, butitalsoextends totheframework offinitequadraticvariation
processes. For a proof we refer to Theorem 2.1 of [39].
Theorem 2.1 Let F : [0,T] R R be of class C1,2([0,T] R) and X = (X )
t t∈[0,T]
× −→ ×
be a real continuous finite quadratic variation process. Then, the following Itˆo’s formula
holds, P-a.s.,
t t
F(t,X ) = F(0,X )+ ∂ F(s,X )ds+ ∂ F(s,X )d−X
t 0 t s x s s
Z0 Z0
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1 t
+ ∂2 F(s,X )d[X] , 0 t T. (2.2)
2 xx s s ≤ ≤
Z0
2.1.1 The deterministic calculus via regularization
A useful particular case of finite dimensional stochastic calculus via regularization arises
when Ω is a singleton, i.e., when the calculus becomes deterministic. In addition, in this
deterministic framework we will make useof thedefinite integral on an interval [a,b], where
a < b are two real numbers. Typically, we will consider a = T or a = t and b = 0.
− −
Westartwithtwoconventions. Bydefault, everyboundedvariation functionf: [a,b]
→
R will be considered as c`adla`g. Moreover, if f: [a,b] R is a c`adla`g function, we extend
→
it naturally to another c`adla`g function f˜on the real line as follows:
f(b), x > b,
f˜(x) = f(x), a x b,
 ≤ ≤

0, x < a.


Definition 2.4 Let f: [a,b] R be aca`dl`ag function and g: [a,b] R be in L1([a,b]).
→ →
(i) Suppose that the following limit
b f(s+ε) f(s)
g(s)d−f(s) := lim g(s) − ds,
Z[a,b] ε→0+Za ε
exists and it is finite. Then, the obtained quantity is denoted by gd−f and called
[a,b]
(deterministic, definite) forward integral of g with respect to f (on [a,b]).
R
(ii) Suppose that the following limit
b f(s) f(s ε)
g(s)d+f(s) := lim g(s) − − ds,
Z[a,b] ε→0+Za ε
exists and it is finite. Then, the obtained quantity is denoted by gd+f and called
[a,b]
(deterministic, definite) backward integral of g with respect to f (on [a,b]).
R
Let us now introduce the deterministic covariation.
Definition 2.5 Let f,g: [a,b] R be continuous functions and suppose that 0 [a,b].
→ ∈
The (deterministic) covariation of f and g (on [a,b]) is defined by
1 x
[f,g](x) = [g,f](x) = lim (f(s+ε) f(s))(g(s+ε) g(s))ds, x [a,b],
ε→0+ ε Z0 − − ∈
if the limit exists and it is finite for every x [a,b]. If f = g, we set [f] := [f,f] and it is
∈
called quadratic variation of f (on [a,b]).
WenoticethatinDefinition2.5thequadraticvariation[f]iscontinuouson[a,b], sincef
is a continuous function. We conclude this subsection with an integration by parts formula
for the deterministic forward and backward integrals.
7
Proposition 2.3 Let f: [a,b] R be a ca`dl`ag function and g: [a,b] R be a bounded
→ →
variation function. Then, the following integration by parts formulae hold:
g(s)d−f(s) = g(b−)f(b) f(s)dg(s) (2.3)
−
Z[a,b] Z]a,b]
and, if f is continuous,
g(s)d+f(s) = g(b)f(b) f(s)dg(s). (2.4)
−
Z[a,b] Z]a,b]
Proof. Letusprovetheintegrationbypartsformula(2.4)relativetothebackwardintegral.
By its definition, we are led to consider the following expression, for ε> 0,
b f(s) f(s ε) 1 b 1 a
g(s) − − ds = g(s+ε)f(s)ds g(s+ε)f(s)ds
ε ε − ε
Za Zb−ε Za−ε
b g(s+ε) g(s)
− f(s)ds.
− ε
Za
Since g(s) = g(b), for s b, we get
≥
1 b 1 b ε→0+
g(s+ε)f(s)ds = g(b) f(s)ds g(b)f(b).
ε ε −→
Zb−ε Zb−ε
Moreover, since f(s)= 0, for s < a, we obtain
1 a
g(s+ε)f(s)ds = 0.
ε
Za−ε
Finally, by Fubini’s theorem we have
b g(s+ε) g(s) b 1
− f(s)ds = dg(r) f(s)ds
ε ε
Za Za (cid:18)Z]s,s+ε] (cid:19)
1 b∧r
= f(s)ds dg(r).
ε
Z]a,b+ε] (cid:18)Za∨(r−ε) (cid:19)
Recalling that g(s) = g(b), for s b, and f(s)= 0, for s < a, we obtain
≥
b g(s+ε) g(s) 1 b∧r
− f(s)ds = f(s)ds dg(r)
ε ε
Za Z]a,b] (cid:18)Za∨(r−ε) (cid:19)
1 r
= f(s)ds dg(r).
ε
Z]a,b] (cid:18)Zr−ε (cid:19)
Sincef iscontinuousatr ]a,b],itfollowsthat r f(s)ds/ε f(r)asε 0+. Therefore,
∈ r−ε → →
by Lebesgue’s dominated convergence theorem, we find
R
1 r ε→0+
f(s)ds dg(r) f(r)dg(r),
ε −→
Z]a,b] (cid:18)Zr−ε (cid:19) Z]a,b]
which implies the validity of (2.4). Theintegration by parts formula relative to theforward
integral is proved analogously. ✷
8
2.2 The spaces C([ T,0]) and C([ T,0[)
− −
Let C([ T,0]) denote the set of real continuous functions on [ T,0], endowed with supre-
− −
mum norm η = sup η(x), for any η C([ T,0]).
k k∞ x∈[−T,0]| | ∈ −
Remark 2.1 We shall develop functional Itˆo calculus via regularization firstly for time-
independent functionals : C([ T,0]) R, since we aim at emphasizing that in our
U − →
framework the time variable and the path play two distinct roles, as emphasized in the
introduction. This, also, allows us to focus only on the definition of horizontal and vertical
derivatives. Clearly, everything can be extended in an obvious way to the time-dependent
case : [0,T] C([ T,0]) R, as we shall illustrate later. ✷
U × − →
Consider a map : C([ T,0]) R. Our aim is to derive a functional Itˆo’s formula
U − →
for . To do this, we are led to define, in the spirit of [16] and [5], the functional (i.e.,
U
horizontal and vertical) derivatives for . Since the definition of functional derivatives
U
necessitates of discontinuous paths, in [5] the idea is to consider functionals defined on the
space of c`adla`g trajectories D([ T,0]). However, we can not, in general, extend in a unique
−
way a functional defined on C([ T,0]) to D([ T,0]). Our idea, instead, is to consider
U − −
an intermediate space between C([ T,0]) and D([ T,0]), denoted C([ T,0]), which is the
− − −
space of bounded trajectories on [ T,0], continuous on [ T,0[ and with possibly a jump
− −
at 0. We endow C([ T,0]) with a (inductive) topology such that C([ T,0]) is dense in
− −
C([ T,0]) with respect to this topology. Therefore, if is continuous with respect to the
− U
topology of C([ T,0]), then it admits a unique continuous extension u: C([ T,0]) R.
− − →
Definition 2.6 We denote by C([ T,0]) the set of bounded functions η: [ T,0] R such
− − →
that η is continuous on [ T,0[, equipped with the topology we now describe.
−
Convergence. We endow C([ T,0]) with a topology inducing the following convergence:
−
(η ) converges to η in C([ T,0]) as n tends to infinity if:
n n
−
(i) η C, for any n N, for some positive constant C independent of n;
n ∞
k k ≤ ∈
(ii) sup η (x) η(x) 0 as n tends to infinity, for any compact set K [ T,0[;
x∈K| n − | → ⊂ −
(iii) η (0) η(0) as n tends to infinity.
n
→
Topology. For each compact K [ T,0[ define the seminorm p on C([ T,0]) by
K
⊂ − −
p (η) = sup η(x) + η(0), η C([ T,0]).
K
| | | | ∀ ∈ −
x∈K
Let M > 0 and C ([ T,0]) be the set of functions in C([ T,0]) which are bounded by
M
− −
M. Still denote p the restriction of p to C ([ T,0]) and consider the topology on
K K M
−
C ([ T,0]) induced by the collection of seminorms (p ) . Then, we endow C([ T,0])
M K K
− −
with the smallest topology (inductive topology) turning all the inclusions i : C ([ T,0])
M M
− →
C([ T,0]) into continuous maps.
−
9
Remark 2.2 (i) Notice that C([ T,0]) is dense in C([ T,0]), when endowed with the
− −
topology of C([ T,0]). As a matter of fact, let η C([ T,0]) and define, for any n
− ∈ − ∈
N 0 ,
\{ }
η(x), T x 1/n,
ϕ (x) = − ≤ ≤ −
n
(n(η(0) η( 1/n))x+η(0), 1/n < x 0.
− − − ≤
Then, we see that ϕ C([ T,0]) and ϕ η in C([ T,0]).
n n
∈ − → −
Now, for any a R define
∈
C ([ T,0]) := η C([ T,0]): η(0) = a ,
a
− { ∈ − }
C ([ T,0]) := η C([ T,0]): η(0) = a .
a
− { ∈ − }
Then, C ([ T,0]) is dense in C ([ T,0]) with respect to the topology of C([ T,0]).
a a
− − −
(ii) We provide two examples of functionals : C([ T,0]) R, continuous with respect to
U − →
the topology of C([ T,0]), and necessarily with respect to the topology of C([ T,0]) (the
− −
proof is straightforward and not reported):
(a) (η) = g(η(t ),...,η(t )), for all η C([ T,0]), with T t < < t 0 and
1 n 1 n
U ∈ − − ≤ ··· ≤
g: Rn R continuous.
→
(b) (η) = ϕ(x)d−η(x), for all η C([ T,0]), with ϕ: [0,T] R a c`adla`g
U [−T,0] ∈ − →
bounded variation function. Concerning this example, keep in mind that, using the
R
integration by parts formula, (η) admits the representation (2.3).
U
Ontheotherhand,considerthefunctional (η) = sup η(x),forallη C([ T,0]). It
x∈[−T,0]
U ∈ −
is obviously continuous, butit is not continuous with respect to the topology of C([ T,0]).
−
As a matter of fact, for any n N consider η C([ T,0]) given by
n
∈ ∈ −
0, T x T ,
− ≤ ≤ −2n
ηn(x) = 2nT+1x+2, −2Tn < x ≤ −2nT+1,
 2n+1x, T < x 0.
− T −2n+1 ≤


Then, (η ) = sup η (x) =1, for any n. However, η converges to the zero function
U n x∈[−T,0] n n
inC([ T,0]),asntendstoinfinity. ThisexamplewillplayanimportantroleinSection3to
−
justify a weaker notion of solution to the path-dependent nonlinear Kolmogorov equation.
✷
To define the functional derivatives, we shall need to separate the “past” from the
“present” of η C([ T,0]). Indeed, roughly speaking, the horizontal derivative calls in
∈ −
the past values of η, namely η(x): x [ T,0[ , while the vertical derivative calls in the
{ ∈ − }
present value of η, namely η(0). To this end, it is useful to introduce the space C([ T,0[).
−
Definition 2.7 WedenotebyC([ T,0[)thesetofboundedcontinuousfunctionsγ: [ T,0[
− − →
R, equipped with the topology we now describe.
Convergence. We endow C([ T,0[) with a topology inducing the following convergence:
−
(γ ) converges to γ in C([ T,0[) as n tends to infinity if:
n n
−
10